Numerical modeling of long flexible fibers in inertial flows.

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Numerical modeling of long flexible fibers in inertial

flows.

Deepak Kunhappan

To cite this version:

Deepak Kunhappan. Numerical modeling of long flexible fibers in inertial flows.. Fluids mechanics [physics.class-ph]. Université Grenoble Alpes, 2018. English. �NNT : 2018GREAI045�. �tel-01877792�

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Pour obtenir le grade de

DOCTEUR DE LA COMMUNAUTE UNIVERSITE

GRENOBLE ALPES

Spécialité : Mécanique des fluides Energétique, Procédés

Arrêté ministériel : 25 mai 2016

Présentée par

KUNHAPPAN Deepak

Thèse dirigée par Guillaume BALARAC, Maître de Conférences, Université Grenoble Alpes

et codirigée par Pierre DUMONT, Professeur, Université de Lyon,

Barthélémy HARTHONG, Maître de Conférences, Université Grenoble Alpes et Bruno CHAREYRE, Maître de Conférences, Université Grenoble Alpes préparée au sein du Laboratoire des Ecoluments, Géophysiques et Industriels et Laboratoire Sols, Solides, Structures

dans l'École Doctorale I-MEP2 - Ingénierie - Matériaux, Mécanique, Environnement, Energétique, Procédés, Production

Modélisation numérique de

l'écoulement suspensions de

fibres souples en régime inertiel

Numerical modeling of long flexible fibers

in inertial flows

Thèse soutenue publiquement le 15 juin 2018, devant le jury composé de :

Monsieur Guillaume BALARAC

Maître de Conférences, Université Grenoble Alpes, Directeur de thèse Monsieur Eric CLIMENT

Professeur, INP-ENSEEIHT, Institut de Mécanique des Fluides de Toulouse, Rapporteur

Monsieur Patrice LAURE

Directeur de Recherche, Centre National de la Recherche Scientifique, Nice, Rapporteur

Monsieur Franck NICOUD

Professeur, Université de Montpellier, Président du Jury

Monsieur Laurent ORGÉAS

Directeur de Recherche, Centre National de la Recherche Scientifique, Grenoble, Examinateur

Monsieur Pierre DUMONT

Professeur, Institut National des Sciences Appliquées es de Lyon, Co-directeur de thèse

Monsieur Barthélémy HARTHONG

Maître de Conférences, Université Grenoble Alpes, Invité

Monsieur Bruno CHAREYRE

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Abstract

A numerical model describing the behavior of flexible fibers under inertial flows was developed by coupling a discrete element solver with a finite volume solver. Each fiber is discretized into several beam segments, such that the fiber can bend, twist and rotate. The equations of the fiber motion were solved using a 2nd _{order accurate explicit scheme (space and time).} The three dimensional Navier-Stokes equations describing the motion of the fluid phase was discretized using a 4th _{th order accurate (space and time) unstructured finite volume scheme.} The coupling between the discrete fiber phase and the continuous fluid phase was obtained by a pseudo immersed boundary method as the hydrodynamic force on the fiber segments were calculated based on analytical expressions. Several hydrodynamic force models were analyzed and their validity and short-comings were identified. For Reynolds numbers (Re) at the inertial regime (10−2 _{≤ Re ≤ 10}2_{, Re defined at the fiber scale), non linear drag force formulations} based on the flow past an infinite cylinder was used. For rigid fibers in creeping flow, the drag force formulation from the slender body theory was used. A per unit length hydrodynamic torque model for the fibers was derived from explicit numerical simulations of shear flow past a high aspect ratio cylinder. The developed model was validated against several experimental studies and analytical theories ranging from the creeping flow regime (for rigid fibers) to inertial regimes. In the creeping flow regime, numerical simulations of semi dilute rigid fiber suspensions in shear were performed.The developed model was able to capture the fiber-fiber hydrodynamic and non-hydrodynamic interactions. The elasto-hydrodynamic interactions at finite Reynolds was validated with against two test cases. In the first test case, the deflection of the free end of a fiber in an uniform flow field was obtained numerically and the results were validated. In the second test case the conformation of long flexible fibers in homogeneous isotropic turbulence was obtained numerically and the results were compared with previous experiments. Two numerical studies were performed to verify the effects of the suspended fibers on carrier phase turbulence and the numerical model was able to reproduce the damping/enhancement phenomena of turbulence in channel and pipe flows as a consequence of the micro-structural evolution of the fibers.

Résumé

Un modèle numérique décrivant le comportement de fibres souples en suspension dans un écou-lement de fluide en régime inertiel a été développé au moyen d’un couplage entre la méthode des éléments discrets et la méthode des volumes finis. Chaque fibre est discrétisée en plusieurs éléments de type poutre permettant de prendre en compte une déformation (flexion, torsion, allongement) et un mouvement de corps rigide. Les équations du mouvement des fibres sont résolues au moyen d’un schéma explicite du second ordre (temps et espace). Le mouvement de la phase fluide est décrit par les équations de Navier-Stokes, qui sont discrétisées et résolues au moyen d’un schéma aux volumes finis non structurés, d’ordre 4 (temps et espace). Le couplage entre la phase solide (discrète) et la phase fluide (continue) est obtenue par une pseudo méthode IBM (Immersed Boundary Method) dans laquelle l’effort hydrodynamique est calculé analyti-quement. Plusieurs modèles de force hydrodynamique issus de la littérature sont analysés et leur validité ainsi que leurs limites sont identifiées. Pour des nombres de Reynolds (Re) correspon-dant au régime inertiel (10−2 _{≤ Re ≤ 10}2_{, Re défini à l’échelle de la fibre), des formulations} non-linéaires de la force hydrodynamique exercée par un écoulement uniforme sur un cylindre infini sont utilisées. Le couplage a aussi été utilisé pour des fibres rigides en écoulement de Stokes, en utilisant l’expression de la force de traînée issue de la théorie des corps élancés (‘slender body theory’). Une expression du moment hydrodynamique par unité de longueur est obtenu à partir de simulations numériques par volumes finis de l’écoulement autour d’un cylindre élancé.

Le modèle développé a été validé par comparaison avec plusieurs résultats expérimentaux et analytiques, du régime de Stokes (pour des fibres rigides) jusqu’aux régimes inertiels. Dans

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le cas du régime de Stokes, des simulations numériques du cisaillement de suspensions de fibres semi-diluées ont été réalisées. Le modèle développé permet de capturer les interactions hydro-dynamiques et non-hydrohydro-dynamiques entre les fibres. Les interactions élasto-hydrohydro-dynamiques pour Re fini ont été validées dans deux cas. Dans le premier cas, la flèche d’une fibre encastrée-libre dans un écoulement uniforme a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Dans le second cas, la conformation de fibres élancées et très déformables dans un écoulement turbulent homogène et isotrope a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Deux études numériques ont été réalisées pour étudier l’effet de la présence de fibres en suspension sur la turbulence au sein du fluide suspensif. Le modèle numérique a permis de reproduire le phénomène de réduction/amplification de la turbulence dans un écoulement en canal ou en conduite, dû à l’évolution microstructurale de la phase fibreuse.

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Acknowledgements

I would like to thank my supervisors (in alphabetical order) : Barthélémy, Bruno, Guillaume and Pierre for providing me the opportunity to pursue this thesis. This thesis had benefited greatly from their continuous valuable sugges-tions and advices. To be specific, Bart for his patience in answering my silly and fundamental questions on solid mechanics and Discrete Element Methods, Bruno for his guidance on the programming parts and working with YADE, Guillaume for teaching me turbulence and occasional help with the YALES2 code and Pierre in providing insights on the physics of fiber suspensions. Spe-cial thanks to Patrick (LEGI) for his advices and ideas on the code coupling, implementation and helping with the HPC facilities. Rémi (3SR) for assisting and solving the computer related issues during my time at 3SR lab. If it wasn’t for these people, this thesis would not have materialized!

I would like to thank my office mates Abdelali and Aleksandr (Sasha) for the frequent entertainment, jokes, coffee breaks and discussions during the tough times. If it was not for these guys I would have lost my sanity long ago (that being said, you guys are still crazy!). Special thanks to Payam for listening to my complaints and cynical comments on almost everything, helping me with paper works and the french bureaucracy, I shall always remain indebted to you! Outside the lab, I would like to thank Keshav for the long thought provoking discussions with coffee and some other things, Hengdi for the weekend dinners, Didier for introducing me to the Metal scene at Grenoble and the weekly guitar jams, Brigitte for the french lessons. I would also like to thank Anish, Harsha, Joe, Vishwa, Mathew (G!), Vedaj and Godson for the frequent inquiries on my well being.

Finally I would like to thank my parents, my brother, my sister and my niece for their unconditional encouragement, love and support and for being with me all the time (I have no words to describe this). Thank you very much!

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CHAPTER

1

INTRODUCTION

Introduction

This thesis is mainly concerned with the numerical modeling of high aspect ratio flexible fibers in fluid flows. These fibers are usually long and thin, i.e, their length is many times greater than their width.

Figure 1.1: Optical micrograph (obtained from Atomic Force Microscopy (AFM)) of cellulose nanofibril suspension adapted from Martoïa et al. [111]

Figure 1.1 shows an optical micrograph of cellulose nanofibril suspension with 0.001% concen-tration by weight. This low concenconcen-tration was used for imaging long cellulose nanofibrils and usually the suspensions are used with 1% volume concentration. From the figure one can notice the complex geometry of these slender fibers The flow of suspensions involving long flexible fibers are quite complex and arise in several industrial applications such as in the manufacturing of paper (pulp fibers) [106], bio based composite materials [50]. In addition to industrial applica-tions, suspensions of long thin filaments are also encountered in several bio-physical phenomena, such as in cell division [118], swimming of microorganisms [90] and sperm cells [125]. In the manufacturing of composite materials involving fibers, the structure of the fiber network and the orientation distribution of the fibers (the microstructure) play a crucial role in the resulting mechanical properties of these composite materials. Figure 1.2from Dumont et al. [51] shows the resulting fiber microstructure of bundle fiber suspensions after compression tests. The fiber

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volume fraction is 0.15 (15%) and two specimens were considered : one with a random initial in-plane (in the e1 direction) microstructure (Ainit11 ≈ 0.5) and oriented in-plane microstructure Ainit

11 ≈ 0.6. The parameter Ainit is the initial fiber orientation tensor. From the figure it is ob-served that the suspension with preferential orientation exhibit lower axial Cauchy stress. The deformed specimens are shown in the right hand side of figure.

Figure 1.2: Fiber microstructure after compression tests of two samples of bundle fiber suspen-sions. Two specimens with initial random (Ainit

11 ≈ 0.5, upper part) and oriented (Ainit11 ≈ 0.6, lower part) microstructures. The graph at the center gives the recorded axial Cauchy stress and logarithmic strains. Adapted from Dumont et al. [51]

Furthermore it is necessary to understand the key mechanisms on the formation of clusters within in fiber networks i.e. floc formation. Another interesting phenomena in these complex flows is the deformation of the suspended fibers, that is there is an elasto-hydrodynamic interaction between the suspended fibers and carrier flow and how the fiber networks in turn affect the flow behavior of the carrier phase.

Objectives

The objective of this thesis is to develop a numerical model through which can one can study the flow of suspensions of fibers for variety of flow situations. In this thesis, the fibers are considered as discrete Lagrangian beams that can deform and the fluid phase is considered as an Eulerian field. The motion and mechanical behaviour of the discrete fiber beams are modeled using an Discrete Element (DEM) solver and the Navier-Stokes equations governing the fluid motion

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are solved using a Finite Volume solver (FVM). The fiber-fiber hydrodynamic and mechanical interactions and the fluid-fiber hydrodynamic force and torque are taken into account in the fiber mechanical model. The resulting hydrodynamic force from the fiber is introduced into the Navier-Stokes equations as line force distribution.

Thesis outline

• Chapter2 introduces the concept of hydrodynamic force and torque on non-spherical par-ticles. The scales of motion in the particle phase and the fluid phase are explained. The Chapter closes with a brief review on the computational methods for modeling particle laden multiphase flows.

• Chapter 3 introduces the solvers used in the present study. This Chapter primarily deals with the description of the numerical methods and schemes used in the present study. The governing equations of the fiber motion and its discretization are presented, followed by a section on the finite volume discretization. Finally the coupling methodology between the solvers developed in this study is presented.

• Discussion on three types of hydrodynamic force models and their shortcomings such as the effect of the fiber discretization and the validity of the force models in the stokesian and inertial regimes are explained in Chapter4. First part of this Chapter presents basic results obtained from a commonly used (the prolate spheroid model) hydrodynamic force model for low Reynolds number flows (at the fiber scale). The remaining part of the Chapter deals with examination of previously derived analytical expressions for the hydrodynamic force for various Reynolds numbers (at the fiber scale). In the last section of the Chapter, a hydrodynamic torque model based on a ‘per unit length’ fashion has been originally formulated from explicit numerical simulations of a slender cylinder in shear flow.

• Chapter 5 consists of validation studies of the coupled numerical method. These studies are performed in the inertial regime (finite Reynolds number at the fiber scale.). The validation tests include :

– Uniform flow past a high aspect ratio fiber with one of its ends fixed with the other end free to bend. The numerically obtained deflection is compared with previous experimental results and an analytical expression.

– The conformation of flexible fibers in homogeneous isotropic turbulent flow. The results are compared with experimental studies.

– Turbulent channel flow of concentrated (volume fraction = 1%) fiber suspension. The numerically obtained flow behavior is compared with experimental results.

• In Chapter 6 numerical study on the rheological behavior of semi-dilute suspensions of rigid fibers are presented. In this Chapter, a Stokesian force model is used to describe the hydrodynamic interaction between the fiber and the fluid. The developed model is validated against available analytical theories and experimental studies hence validating the developed coupled numerical model at low Reynolds numbers.

• Chapter 7 deals with the numerical study of turbulent pipe flow of semi dilute fiber sus-pensions. Preliminary qualitative results on the drag reducing effects the suspension are presented

• Chapter8gives an overall conclusion of the present work, discussing possible improvements and drawbacks of the developed coupled numerical model.

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CHAPTER

2

REVIEW OF COMPUTATIONAL METHODS FOR PARTICULATE

MULTIPHASE FLOWS

Contents

2.1 Introduction . . . 5

2.2 Microhydrodynamics . . . 5

2.2.1 Resistance and Mobility Tensors . . . 8

2.2.2 Higher Reynolds numbers . . . 9

2.3 Computational modelling of particulate multiphase flows . . . 11

2.3.1 Simulation of the fluid phase . . . 11

2.3.2 Simulation of particle laden flows . . . 13

2.4 Numerical modeling of rigid and flexible fibers . . . 16

2.5 Objectives of the present work . . . 18

2.1 Introduction

The present Chapter gives an introduction to the field of particulate multiphase flow in the context of fiber suspension flows. Some important results and historical development in the field of microhydrodynamics are presented, in particular the drag force on spherical and spheroidal particles in Stokes flows. The properties, theorems and solution methods to Stokes flow problems have been omitted and for such details, the reader is advised to see the following references [88,

66,64]. In the next part, popular methods used for the numerical study of multiphase particulate flows are presented. The Chapter closes with discussion on the modelling of fiber suspensions and rheological and microstructural behaviour of rigid non Brownian fiber suspensions.

2.2 Microhydrodynamics

The motion of a particle suspended in a quiescent fluid is governed by the hydrodynamic force exerted by the fluid on the particle boundaries, the equation of motion for an incompressible fluid (Navier-Stokes equations) reads as

∇ · ~Uf = 0 (2.1a)

where in equation (2.1a) is the incompressibility constraint, ~Uf is the fluid velocity. The mo-mentum equation reads as

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∂ ~Uf ∂t + ~∇ ~Uf ⊗ ~Uf =₋∇P~ ρ + ν ~∇ 2_U_~ f (2.1b)

where, ρ is the fluid density, p the pressure and ν the kinematic viscosity of the fluid.

In order to characterize the effects of particle inertia suspended in the fluid, two non-dimensional numbers viz. the Stokes (St) and particle Reynolds number (Rep) are frequently used. The Stokes number is defined as

St = |~Uf|τp lp

, (2.2)

where in equation (2.2) lp is the characteristic length scale of the particle and τp is the particle relaxation time:

τp ∝ ρplp2

µf

. (2.3)

The Reynolds number is defined as

Rep = |Uf|lp

ν . (2.4)

The Reynolds number is the ratio of inertial forces to viscous forces, and the Stokes number is defined as the ratio of characteristic time scale of the suspended particle to the characteristic time scale of the fluid flow. For low Stokes numbers (St << 1), the suspended particle follow the streamlines of the flow and for higher St, the particles detach from the flow and do not necessarily follow the fluid streamlines. When the particle Reynolds number is very small (Re << 1) the equation (2.1b) can be simplified by neglecting the convective and transient part to obtain linear Stokes equation, hence

~ ∇P

ρ = ν ~∇ 2_U_~

f (2.5)

Since the Stokes equations are linear (2.5), it is possible to derive analytical solutions by the use of several mathematical methods such as Green’s functions [80,66] or by boundary integral methods [186] and singularity methods [131]. Stokes [159] derived the frictional resistance or the ‘drag’ force experienced by a sphere having radius r moving in a quiescent fluid, the expression reads as

~

Fd= 6πµr ~Uf. (2.6)

For a rigid spherical particle suspended in an unbounded Newtonian quiescent fluid, Faxén [57] derived general expressions for the force and torque relating to the particle translational and rotational velocity ~ Fhyd = 6πµr 1 +r 2 6 ∇~ 2 ~u0_{− (~}U _{− ~}U_f∞) (2.7a) ~ Thyd = 8πµr3 1 2 ~∇ × ~u 0 −~Ω − ~Ω∞f (2.7b) where in equations (2.7a) and (2.7b), ~u0 _{is the disturbance velocity caused by the particle.}

Batchelor [14] extended Faxén’s law by the concept of stresslet. The stresslet S is the symmetric part of the first moment of force [64]. The stresslet relates the stress induced by the particles in the suspesnion [64], it is in fact the resistance of the particle deformation in strainig motion. The equation for the stresslet for a sphere is:

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2.2 Microhydrodynamics S = 10 3 2E∞+ 1 + 1 10r 2_∇_~2 (~_∇~u0 + (~_∇~u0)T₎ . (2.8)

Jeffery [70] derived equations of motion for a neutrally buoyant ellipsoid in an unbounded stokes flow. For an ellipsoid having b as the length of the minor axis, and a as the semi-major axis, with aspect ratio rp defined as a/b, the time period of rotation for the ellipsoid immersed in a shear flow at Re → 0 is

tp = 2π ˙γ

rp+_r1_p (2.9)

Figure 2.1: Rotation of a spheroidal object in shear flow (shear in XY plane).

In equation (2.9), ˙γ is the shear rate (see figure2.1). Considering ~p being the orientation vector of the symmetric axis of the ellipsoid, φ and θ being the angles ~p makes with velocity gradient direction and vorticity direction (see fig. 2.1), the expressions for the angular velocity of the ellipsoid is dφ dt = ˙γ r2 p+ 1 (r2_pcos2φ + sin2φ) (2.10) dθ dt = ˙γ 4 _r2 p− 1 r2 p+ 1 sin 2θ sin 2φ (2.11)

with the angles φ and θ given as

tan φ = rptan ˙γt rp+_r1_p + k (2.12) tan θ = q Crp r2 pcos2φ + sin2φ (2.13) where C and k are constants depending upon the initial orientation of the ellipsoid.

In addition to the motion of the ellipsoid in a quiscent fluid, Jeffery derived an expression for the effective viscosity for a dilute suspension of spheroidal particles largely influenced from Einstein’s work [56]. Concerning cylindrical particles, Burgers [30] derived the hydrodynamic force and torque constants on a cylindrical particle, drawing inspiration from the work of Oberbeck [120].

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Burgers had postulated that the disturbance induced by a long ellipsoid of revolution could be represented as a line force of magnitude acting along the axis of symmetry of the particle. Broersma [26,27] improved the work of Burgers [30], however the results of both Burgers and Broersma were not asymptotically accurate. Mason and co-workers [167, 112] experimentally showed that neutrally buoyant rigid rod like particles tend to follow orbits similar to Jeffery orbits and in fact the expression (2.9)- (2.13) could be used for the prediction of the motion provided that the aspect ratio of the rod rp is replaced by an effective aspect ratio re, (re = 0.8rp). Bretherton [24] theoretically showed that the motion of axisymmetrical particles would exhibhit motion similar to Jeffery’s predictions. Cox [41] by the method of Matched Asymptotic Expansions (MAE) derived the transverse and axial force per unit length on a high aspect aspect ratio cylindrical particle known as the slender body theory. Batchelor [12] extended the formualtion of Cox for slender bodies with non-circular cross sections. The slender body theory was further improved by Keller and Rubinow [77] and later Johnson [71] introduced a non local drag formulation in the slender body theory.

2.2.1 Resistance and Mobility Tensors

The solution to microhydrodynamics problems can be classified into two types : the resistance problem and the mobility problem. In the resistance problem, the fluid and particle velocities are set as the boundary conditions and one seeks for the hydrodynamic force, torque and stresslet the fluid applies on the particle [23].

  ~ Fhyd ~ Thyd Shyd  = µ   A B G B C H G H M  ·   ~ Uf − ~Up ~ Ωf − ~Ωp E_∞   (2.14)

where in equation (2.14) A, B, and C are second rank tensors, G and H are third order tensors and M is a fourth order tensor, µ is the fluid dynamic viscosity. The square matrix in (2.14) is usually known as the ‘grand resistance’ matrix. Expanding the equation (2.14), leads to the following set of expressions [23]. The expression of the hydrodynamic force on the particle reads as Fhydi = µ h Aij U_jf _{− U}_jp+ ˜Bij Ωf_j _{− Ω}p_j+ GijkEjk∞ i (2.15) The expression for the hydrodynamic torque reads as:

Thydi= µ h Bij U_jf− Ujp + Cij Ωf_j − Ωpj + HijkEjk∞ i (2.16) and the expression for the particle stresslet is

Shydij = µ h Gijk U_kf− Ukp + Hijk Ωf_k− Ωpk + MijklE∞kl i (2.17) The tensors appearing in equations (2.14) to (2.17) are called the hydrodynamic resistance ten-sors. The hydrodynamic resistance tensors can be derived for any particle and it primarily depends upon the particle size and shape. The derivation of resistance tensors has been pre-sented in the works of Brenner. In ref [21, 22], Brenner presents the method of derivation and in ref [23], the hydrodynamic resistance tensors of various slender axissymmetric particles have been presented. Furthermore, analytical expressions for the specific viscosity, rotary diffusion and other microstructural properties of Newtonian suspensions (dilute to semi-dilute) regime has been reviewed.

The mobility problem aims to derive the solution of the particle motion in response to prescribed forces and torques under an ambient flow. Based on the linearity of the Stokes equations, equation (2.14) can be written as

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2.2 Microhydrodynamics   ~ Uf − ~Up ~ Ωf − ~Ωp µ−1S  =   a b g b c h g h m  ·   µ−1F~hyd µ−1T~hyd E∞   (2.18)

where in equation (2.18), a, b and c are second order tensors. g and h are third order tensors. m is a fourth order tensor. The square matrix in the righjt hand side of (2.18) is called the mobility matrix. The components of the mobility matrix can be derived from the resistance matrix, for details see [80] pages 109-115.

2.2.2 Higher Reynolds numbers

The formulation of hydrodynamic forces and torques presented in the previous section are only valid in the Stokes regime. Furthermore, the drag force on an infinite cylinder in the limit of low Reynolds numbers (Re → 0) cannot be derived using the Stokes equations (Stokes paradox) [88]. In this section we discuss some of the analytical and empirical expressions for the hydrodynamic forces and torques on spherical and non-spherical particles when the particle Reynolds numbers are in the inertial regime. Oseen [121] derived an expression for the hydrodynamic force on a sphere for finite inertia. The solution was derived by linearizing the convective part of the NS equations, Ux1 ∂ui ∂x1 =₋1 ρ ∂p ∂xi + ν ∂ 2_u i ∂xj∂xj (2.19)

where in eqaution (2.19) Ux1 is the uniform velocity in the stream-wise direction (x1), far from

the particle in an unbounded domain, ~u is the disturbance in the flow velocity induced by the particle. The boundary conditions for the problem are

At the particle surface, the boundary condition reads:

u = U (2.20a)

and at the far field, the boundary conditions are:

u→ 0 and (2.20b)

p_{→ p}_∞ for r → ∞ (2.20c)

where in equation (2.20c), r is the distance from the particle and p∞ is the free stream pressure. The expression of the drag force on a sphere was derived as [121]

~ Fhyd = 6πµr ~U 1 + 3 16Re (2.21) The Oseen’s equations (2.19) and (2.20) are not entirely correct. Due to the linearization of the NS eqautions and the no-slip boundary condition on the surface, the inertial terms tend to zero at the particle surface. At some distances away from the particle, The inertial forces dominate and at the farfield viscous terms tend to zero, which implies that neither the Stokes equations or the Oseen equations are valid throughout the domain, this is called the Whitehead paradox. Nevertheless Oseen’s linearization technique has been widely used to study problems in microhydrodynamics at finite Reynolds numbers. With the method of Matched Asymptotic Expansions (MAE), Proudman and Pearson [133] derived exact analytical expression on the drag force of a sphere with more accuracy, their expression reads as

~ Fhyd= 6πµr~u 1 + 3 16Re + 9Re2_ln(Re/2) 160 + O(Re 2₎ (2.22)

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The method of MAE has been extensively used to derive the drag force per unit length on an infinite cylinder by Proudman and Pearson [133], Kaplun and Lagerstrom [75,74] and Skinner [153]. Hydrodynamic drag and lift forces for flow past infinite cylinders have been derived semi-empirically for higher Reynolds numbers by Taylor [163]. Taylor [163] used these results for analyzing the swimming and motion of long, axisymmetric animals [163]. Similar results for low Reynolds Re ≈ O(10−3₎ _{was derived by Tomotika et al. [}₁₆₄_{]. Breif details on the} derivation of these solutions are presented in Chapter4. Junk and Illner [73] re-derived Jeffery’s equation based on asymptotic expansions and the solution was extended to particles of general shape and finite Reynolds numbers. Recent studies by Einarsson et al. [55] and Rosen [138], analyzed the motion of spheroidal and ellipsoidal particles in simple shear flows for finite shear Reynolds numbers. Their study was primarily based on perturbation and expansion methods complemented by Direct Numerical Simulation (DNS) reuslts using Lattice Boltzmann methods. In the work of Einarsson et al., the orientation motion of neutrally buoyant spheroids were studied. The effect of inertia on the Jeffery orbits and the mechanics of particle rolling and tumbling were studied. In the work of Rosen et al. [138] the effect of inertia on the motion of triaxial ellipsoids were analyzed and studied. Several drag models for non-spherical particles in higher Reynolds number have been presented and most of these drag models are semi-empirical derived from experiments and Direct Numerical Simulations (DNS) (see section 2.3). Clift et al. [39] presented various drag models for non-spherical particles with short aspect ratios for various particle Reynolds numbers, these models were composed of analytical models for low Reynolds numbers and models from experimental fitted data. Holzer and Sommerfeld [67] gave the following expression for the drag coefficient for a nonspherical particle

CD = 8 Re 1 pφ_k + 16 Re 1 √ φ+ 3 √ Re 1 φ34 + 0.42100.4(−logφ)0.2 1 φ_⊥ (2.23)

where in equation (2.23) CD is the drag coefficient, defined as CD = | ~

FD| 0.5ρ_|~U_|2 πd2

4

(2.24) The expression of the drag coefficient presented in equation (2.23) includes a term φ called the sphericity of the particle. Sphericity of a non-spherical particle is defined as the ratio of the surface of the volume equivalent sphere to the surface area of the non-spherical particle. The parameter φ⊥ is the cross-wise sphericity, defined as the ratio of the cross-sectional area of the volume equivalent sphere to the projected cross-sectional area of the non-spherical particle perpendicular to the flow direction. The parameter φ_k is the lengthwise sphericity defined as the ratio of the cross-sectional area of the volume equivalent sphere to the cross-sectional area of the non-spherical particle projected parallel to the flow direction. This model of the drag coefficient was developed by combining the correlations obtained from the experimental studies in refs. [93,60,166].

Figure 2.2 shows the variation of the drag-coefficient CD with respect to Reynolds number for various Reynolds numbers from the experimental data of [93, 60, 166]. Drag coefficients of non-spherical particles have also been obtained from DNS. Zastawny et al. [189] performed direct numerical simulations for ellipsoids, disc and fiber shapes. From these numerical sim-ulations particle drag, lift and moment coefficients were derived. Holzer and Sommerfeld [68] also presented the drag and lift coefficients of nonspherical particles by DNS using the Lattice Boltzmann Method (LBM). Most of the drag models are valid for short aspect ratio particles and as the sphericity of the particle tends to lower values, i,e. as the particles approach slender shapes, these drag models do not perform well.

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2.3 Computational modelling of particulate multiphase flows

Figure 2.2: Drag coefficient vs. Reynolds numbers for various particle shapes, from [68].

2.3 Computational modelling of particulate multiphase flows

In this section some computational approaches pertaining to the numerical simulations of par-ticulate multiphase flows are presented. The discussion is restricted to Eulerian-Lagrangian approaches or pure Lagrangian methods. The Eulerian approach for describing the dispersed phase is not considered in the present work. An overview of several methods namely Stokesian dynamics, Lagrangian point force method, Force Coupling Method (FCM) and Fully Resolved Lagrangian methods are presented.

Figure 2.3 shows a ‘map’ of the particle concentration with respect to the coupling approach. In the case of dilute suspensions, the fluid flow affects the particle motion and the momen-tum exchange from the dispersed particle phase to the fluid is not considered. The particle Reynolds numbers are usually in the creeping regime and the particles do not detach from the flow streamlines. In the two-way coupled approach, the dispersed particulate phase affects the carrier flow, however the particle concentration is so low, that the particle-particle hydrodynamic and non-hydrodynamic interactions are negligible. When the particle concentration increases, the particle-particle hydrodynamic interactions become dominant, these hydrodynamic interac-tions could be short range such as lubrication forces between particles or long range such as when the motion of a given particle would be affected by the wake of another particle. Understanding the particle and fluid length and time-scales are of importance when it comes to modelling of particulate flows. Based on the flow Reynolds number, the fluid would exhibit a range of length scales and time scales of motion.

2.3.1 Simulation of the fluid phase

In the case of turbulent flows, the fluid exhibits varieties of length and time scales accompanied by the transfer of energy between the scales (Figure 2.4). Full Resolution of the Navier Stokes equations [Direct Numerical Simulations (DNS)] for turbulent flows remain a challenge to this day as the number of grid points required scales to almost the cube of Reynolds number. Large Eddy Simulations (LES) resolves the larger scales of the fluid motion and the smaller scales are modelled. This is achieved based on temporal and spatial averaging by means of a ‘filter’ which

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Figure 2.3: Particle concentration and fluid-particle coupling. By Loth et al. [104]. removes the information of the small scales. The subgrid scales and the associated turbulent stresses are then modelled using eddy viscosity models such as Smagorinsky model (constant eddy viscosity) [155] or dynamic eddy viscosity models [61].

Figure 2.4: Turbulent structures in a mixing layer flow from Brown and Roshko [29]. 1

Another approach of simulating turbulent flows is based on the Reynolds Averaged Navier Stokes equations (RANS). These equations are the time averaged Navier Stokes equations, in which the instantaneous velocity is decomposed into a mean velocity field and the fluctuating velocity

1_{Adapted from a report by U.Piomelli titled ‘A primer on DNS and LES’.}

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Ener gy T

rans fer

Energy containing large

eddies Dissipating eddies

energy dissipation

Increasing Grid Resolution -->

Large Eddy Simulation (LES) Direct Numerical Simulation (DNS)

Resolved

Modeled

Reynolds Averaged Navier Stokes (RANS)

Resolved

_Modeled

Figure 2.5: Comparison of resolution requirements in various turbulence simulation approaches. ∆x refers to the grid spacing.

field. In RANS modeling, one seeks for the solution of the mean velocity field and the stresses induced by the fluctuating velocity is modeled. Several RANS models have been developed and most of these RANS models involve additional transport equations for solving the turbulent kinetic energy/turbulent viscosity or in the case of the seven equation Reynolds Stress Model (RSM) six individual components of the turbulent stresses are solved and one transport equation for the turbulent dissipation is solved. RANS models involve several semi-empirical constants derived from experimental studies and boundary layer theory [178]. Figure2.5shows the various turbulence modelling approaches and the associated grid requirements.

2.3.2 Simulation of particle laden flows

Stokesian Dynamics (SD) is a simulation framework initially used for rigid spherical suspensions for both dilute and dense suspensions [20] similar to Molecular Dynamics methods. The key idea in Stokesian Dynamics is to construct resistance matrices (described in section 2.2.1, equation (2.14)) for the given particle configuration in space. The method is capable of taking into account the non-hydrodynamic interparticle interactions such as Brownian forces and colloidal forces. Swan and Brady extended the method of SD to include particle wall interactions [162]. Typical SD simulations have the computational complexity of O N3

where N is the number of spheres. Sierou and Brady [150] introduced an efficient SD method which reduces the computational complexity to O(N logN). Various studies on colloidal and non-colloidal suspensions based on the SD method can be found in the following refs. [84,19,151]

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The Lagrangian point force approach is one of the most commonly used methods and has been in use since the late 80s. In this method, the motion of individual particles or cloud of particles are tracked in the simulation domain and the fluid velocities from the Eulerian grid are interpolated to the particle position. The drag force on the particle is calculated by the Stokes drag equation and the motion of the particle is calculated based on Newton’s second law as

md~Up

dt = ~Fd. (2.25)

Yeung and Pope [184, 185] presented a Lagrangian particle tracking algorithm for inertialess particles in isotropic turbulence with respect to spectral methods. This method was based on one-way coupling between the dispersed phase and carrier phase and it was used to study the Lagrangian statistics of velocity, acceleration and dissipation in isotropic flow fields. Eaton [52] presented a critical review on the point-particle approach within the context of particle laden flows. A detailed comparison between several experimental and numerical results were also pre-sented. From the comparison between the numerical and experimental studies, it was shown that the Lagrangian point force method was able to reproduce the experimental results in one-way coupled simulations with good agreement. However, for two-way coupled simulations, the numer-ical results differed significantly from the experimental ones especially for the studies involving higher particle Reynolds numbers (Rep ≈ 100) and high Stokes numbers. This discrepancy was due to the fact that at higher particle Reynolds numbers, the effect of the particle unsteady wake is significant and the point force methods are not able to describe these features very well and little to no turbulence modulation was observed in the numerical studies. Balachandar and Eaton [11] presented a critical review on various approaches for modelling turbulent dispersed flows.

Figure 2.6: Applicability of various coupling approaches from [11]. τk is the Kolmogorov time scale, the time scale of the smallest eddies in the carrier flow and η is the smallest length scale (Kolmogorov length scale). d is the particle length scale

Figure 2.6shows a plot of the time scale ratio (τp/τk) versus the particle size ratio (d/ηk) and presents the applicability of various coupling approaches. The particle relaxation time or time

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scale is represented by τp and τkis the Kolmogorov time scale, i,e. the time scale of the smallest eddies. d is the particle length scale and η is the Kolmogorov length scale.

The Force Coupling Method (FCM) [40,102] is another approach to model the coupling between the carrier phase and the dispersed phase. In this method, the presence of the particle is repre-sented by a force distribution in the NS equations. This body force is reprerepre-sented as multipole expansion of the particle force [102]:

fi(x, t) = Np X n=1 Fi∆ (x− Yn(t)) + Np X n=1 Gn_ij ∂ ∂xj ∆0(x− Yn(t)) (2.26)

where in equation (2.26), the first term on the RHS Fi is the force monopole, that the particle exerts on the fluid which is comprised of the inter-particle interaction forces, particle inertia and other external forces. Yn_(t) _{is the position of the particle at time t. The second term in the} RHS Gij is the force dipole related to the moment of the forces acting on the fluid, consisting of symmetric part and an anti-symmetric part. The anti-symmetric part corresponds to the hydrodynamic torque and the symmetric part corresponding to the stresslet. The information about the particle size enters equation (2.26) via the Gaussian envelopes ∆(x) for the monopole and ∆0

(x) for the force dipole as

∆(x) = (2πσ2)−3/2exp −_2σ|x|₂ (2.27a) and ∆0(x) = (2πσ02)−3/2exp − |x| 2σ02 (2.27b) where σ = a/p(π) and σ0 _{= a/(6}√_π)1/3 _{, a is the particle radius.}

The FCM method introduces a spatially filtered velocity which leads to higher accuracy compared to conventional point force approaches especially when the turbulence in the carrier medium is resolved by Direct Numerical Simulations [11]. In fact FCM is nearly exact for low particle Reynolds numbers ≈ O(10) and has been extensively used to study turbulence modulation by spherical particles in channel flows [101], turbulent shear flows [175], sedimentation of spherical particles in Stokes flows [114] and micro-swimmer suspensions [45,46].

Glowinski et al. [62] developed a fictitious domain method known as the Distributed La-grangian Multiplier method (DLM/FD). The equations of fluid and particle motions are com-bined into a weak formulation by a ‘comcom-bined velocity space’ (page 766 in [62]). The fluid flow equations are enforced inside the particle constrained to a rigid body motion by means of Lagrange multipliers. The multiplier represents an additional body force per unit volume to maintain the rigid body motion in the interior of the particle. The equations are discretized based on a finite element framework. A ’fine’ mesh is used to solve the fluid velocity and a coarser mesh is used for the pressure field. The particle is represented by another finite element mesh. Time advancement of the combined system is performed based on operator splitting tech-niques (Marchuk-Yamenko) similar to the method of Chorin [35]. The Lagrange multipliers are used as force densities in the last step of the fractional step of the time advancement. Singh et al. [152] and Patankar et al. [124] presented a modified version of the DLM method, with improved collision physics between the particles similar to a soft sphere approach, where the particles can overlap each other. Within the context of spectral finite element methods, Dong et al. [48] presented a DLM method. Studies on settling of spheres under gravity in a fluid column has been published using this method [152,62,187].

Immersed Boundary methods (IBM) and its variants have been popular in simulating par-ticulate multiphase flows. Peskin [126,128,127] presented one of the first IBM approaches for studying the fluid dynamics of heart valves. In this method, the NS equations are solved on a fixed cartesian grid and the discretization of the NS equations can be based on finite difference

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methods/volumes or elements and the solid object is represented by discrete Lagrangian marker ‘fibers’. The relative displacement of these lagrangian markers by the fluid velocity is used to calculate the elastic response of the fibers. The no slip boundary condition on these marker points is enforced by setting the velocity of the marker points to the velocity of the fluid in the grid points in the vicinity of the marker. Based on Peskin’s IB method, Uhlmann [169] devel-oped a direct forcing IB formulation. This formulation utilizes the regularized delta functions of Peskin [127] to interpolate the fluid velocity to the Lagrangian marker points and in ‘spreading’ the feed back force from the particle to the fluid grid. The key idea in Uhlmann’s method [169] is to evaluate a body force term in the Lagrangian marker points at an intermediate time level between tn _{and t}n+1 _: ~ Fn+1/2= U~ d_{− ~}_Un ∆t − RHS n+1/2 _(2.28a)

where in equation (2.28a), ~Ud_{is the velocity at the Lagrangian marker, RHS is the right hand side} of the NS equations including the viscous terms and the pressure gradient term. The expression for ~Ud_reads:

~

Ud(X_li) = ~ui_c+ ~ωi_c_{× X}_li_{− x}i_c

(2.28b) where xc is the geometric center of the particle, ~uic the velocity at the center of the particle and ~

ωi

c the rotational velocity of the particle center. The velocity ~Un is interpolated from the fluid grid using the following expression:

~ Un=

X

~ufδh(xf− Xl) h3 (2.28c)

in equation δh is a discrete dirac-delta function and h is the grid spacing. Details of this function can be found in the work of Roma et al. [136]. The feed back force is then interpolated back to the fluid grid using an expression similar to (2.28c). Breugem [25] improved the IB framework of Uhlmann [169] by including a multidirect forcing scheme for better approximation of the no-slip boundary condition on the particle interface. Additional improvements were made in reducing the grid dependency on the IB scheme (‘estimation of the effective particle diameter’) and the overall numerical stability of the IB scheme was improved. The aforementioned IB scheme was modified by Ardekani et al. [8, 9] by including particle-particle interactions for various particle shapes (ellipsoids, spheroids). The method was used to study the influence of particle concentration and shape in drag reducing turbulent flows.

Within context of Lattice Boltzmann Methods (LBM), the ‘standard bounce back’ model and the ‘external boundary forcing methods’ were applied to study particle laden flows. A detailed review has been presented by Aidun and Clausen [2] and a detailed method for the simulation of suspensions using the LB framework has been derived by Ladd [85, 86]. A recent review on numerical methods for the simulation for particle laden multiphase methods has been presented by Maxey [113]. The review covers several other methods such as the Arbitrary Lagrangian Eulerian (ALE) approach and the PHYSALIS methods [190,149] for the simulation of multiphase flows.

2.4 Numerical modeling of rigid and flexible fibers

Rigid fibers Based on the point force coupling approach, several studies on the orientation behavior and preferential concentration in turbulent channel flows for various Stokes numbers and fiber aspect ratios have been presented in refs [33,108,192,191,119]. In these studies, the fibers were modeled as Lagrangian points, utilizing the hydrodynamic force and torque models

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2.4 Numerical modeling of rigid and flexible fibers

based on Jeffery’s theory [70]. The relaxation time of the fiber was calculated using the expression derived from the experimental studies of Shapiro and Goldenberg [145],

τp = 2Da2lnrp+ q r2 p− 1 9νqr2 p− 1 (2.29a) In equation (2.29a) D is the ratio of particle density to fluid density, rp the aspect ratio. The associated Stokes number is

St = τpu 2 τ

ν (2.29b)

where uτ is the turbulent friction velocity. Shin and Koch [147] used the slender body theory of Khayat and Cox [79] valid at finite Reynolds numbers (Reynolds based on the fiber length) for studying the rotational diffusion of long rigid fibers in isotropic turbulence. The integral equations were solved using a pseudo-spectral method [148]. A similar method was used by Lopez and Guazzelli [103] for studying the settling of fibers under a vortical flow. Saintillan et al. [140] used particle mesh Ewald summation algorithm to simulate settling of fiber suspensions. Flexible fibers: In the framework of simulating flexible fibers, Tornberg and Shelley [165] derived a non-local SBT (slender body theory) formulation [77, 71] coupled with the Euler-Bernoulli beam theory to yield integral expressions for the center-line velocity of high aspect ratio filaments. Their numerical model included the hydrodynamic interactions between multiple fibers and the effect of fibers on the background fluid. Li et al [94] used a variation of the numerical method proposed by Tornberg and Shelley [165] to study the sedimentation of flexible filaments. A detailed review of SBT concerning modeling fibers has been presented by Lindner and Shelley [96]. Stockie and Green [158] used Peksin’s method to simulate the dynamics of a single flexible fiber and they were able to reproduce the deformed fiber shapes observed experimentally by Forgac & Mason [58]. Wiens and Stockie [177] developed a parallel scalable IBM based on a Kirchoff’s rod model proposed by Lim et et al. [95]. Aidun [180, 181,141] and coworkers used an IBM within the LBM framework known as 'external boundary forcing' method to study the effect of fiber stiffness and rotational diffusion of semi-dilute sheared fiber suspensions. Immersed Boundary Methods/Fictitious Domain Methods, aim to fully resolve the flow-field around a fiber and are therefore computationally expensive.

In bead/rod models, a fiber is represented by an array of spheres or prolate spheroids. Yamamoto & Matsuoka [183] developed a flexible fiber mechanical model where a fiber consisted of spheres that are bonded to each other. The mechanical properties of the fiber were determined based on the bending, twisting and stretching constants. Their model was validated against low Reynolds number flows such as the motion of a single fiber in shear flow with respect to Jeffery’s predictions [70] and to the deformation of fibers observed experimentally by Forgacs & Mason [58]. Their study only considered one way coupling, ignoring the action of fibers on the fluid. Joung et al. [72] developed a similar model where a fiber was represented by a chain of spheres with inextensible connectors thereby allowing the bending and twisting dynamics of the fibers and the model was used to study the behavior of Newtonian suspensions and predict the viscosity of a flexible fiber suspension. Delmotte et al. [44] improved the model of Yamamoto and Matsuoka [183] by implementing Lagrange multipliers and a new contact model between the spheres known as the Gears model to define the twisting and bending behavior. The model was validated with Jeffery’s predictions, [70] sedimentation of a flexible fiber, dynamics of an actuated filament and swimming of microorganisms.

Schmid et al. [144] developed a model for flexible fibers in which the hydrodynamic force of the fiber was represented as a chain of prolate spheroids (hereby referred to as PS model) and included the contact forces between the fibers. Their model was used to study the formation of flocs in fiber suspensions, effect of fiber equilibrium shapes on the viscosity of fiber suspensions, however their simulations were still based on one-way coupling. Lindström and Uesaka [97] improved

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the numerical stability of the model proposed by Schmid et al. [144] and also implemented a two-way coupling scheme. Moreover a hydrodynamic force model for the inertial regime was proposed. The model was used for the study of semi-dilute non-brownian fiber suspensions [98]. The same model of Lindström & Uesaka [97] was used by Andrić et al. [6,5,7] for studying the behavior of dilute suspensions and motion of fibers in inertial flows. However the hydrodynamic force presented by Lindström & Uesaka [97] are not accurate for intermediate inertial Reynolds numbers and for low angles of attack past the fiber, the component of force parallel to the fiber symmetric axis was neglected. A detailed review of such models have been presented by Lindner and Shelley [96] and Hämäläinen et al. [65].

2.5 Objectives of the present work

In this Chapter, an introduction to particulate multiphase flows have been presented. The formulation of hydrodynamic forces and torques on non spherical particles from the creeping regime to inertial regimes have been explained, followed by short descriptions on computational methods to simulate particulate flows have been presented. From the discussions presented in this Chapter, it is evident that the behavior of dilute and semi-dilute suspensions in inertial flows are not fully understood. This thesis aims at developing a Eulerian Lagrangian method to numerically simulate the collective behavior of several thousands of fibers in creeping and inertial flows. The fluid phase is solved by Finite Volume Method (FVM) and the motion of the fibers are solved based on the Discrete Element Method (DEM). The coupling between the phases is based on a ‘pseudo’ immersed boundary method, as the fibers are discretized into several segments and the length of each segment is comparable to the cell edge length of the fluid domain. The drag force on the fiber segments are obtained via analytical expressions and drag correlation formulae. In Chapter 3, the governing equations of motion and the numerical methods for the fluid and fiber are discussed. Chapter4 presents an analysis on the use of hydrodynamic force and torque models relevant to long fibers (fibers with aspect ratio rp > 10). In Chapter 5 the developed numerical framework is tested for fibers in inertial flows. Chapter 6deals with fibers in the Stokes regime, i.e. a numerical study on semi dilute rigid fiber suspensions in shear is presented. This study highlights the ability of the developed numerical model to reasonably capture and describe the many body hydrodynamic interactions between the fibers. In Chapter

7, preliminary results of the drag reducing effects in semi-dilute fiber suspensions in turbulent pipe flows are presented.

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CHAPTER

3

NUMERICS AND COUPLING METHODOLOGY

Contents

3.1 Introduction . . . 19 3.2 Discrete element solver . . . 20 3.2.1 Fiber mechanical model . . . 21 3.2.2 Numerical stability . . . 27 3.3 Finite volume solver . . . 27 3.3.1 Finite Volume discretization. . . 27 3.3.2 Pressure correction . . . 29 3.3.3 Time advancement . . . 30 3.3.4 Linear solver . . . 30 3.4 FVM-DEM Coupling . . . 30 3.4.1 Overview of the hydrodynamic force and torque on a fiber segment. . . 31 3.4.2 Coupling Methodology . . . 32 3.4.3 Implementation. . . 35 3.4.4 Coupled FVM-DEM numerical stability . . . 37 3.4.5 Coupling timescale . . . 38 3.5 Conclusion . . . 39

3.1 Introduction

This chapter describes the governing equations for the fiber and fluid motions, and the momentum exchange between the two phases. Two separate solvers are used to solve the equations of motion of the fibers and the fluid. The equations of motion of the fibers are solved using the open source Discrete Element Method (DEM) code YADE. The fibers are discretized into several beam segments, with each segment being composed of a pair of nodes (fig. 8.2). The equations of motion for the nodes are solved by a 2nd _{order explicit accurate scheme (time and space).} The high-fidelity finite volume code YALES2 is used to solve three dimensional incompressible Navier-Stokes equations. The fluid domain is discretized using a 4th _{order accurate (space and} time) explicit finite volume scheme. Momentum exchange between the phases is included via the source term in the Navier-Stokes equations.

This chapter begins with a brief outline of the discrete element method and the solver. It is followed by a description of the mechanical model of the fiber and its discretization. An outline

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of the fluid solver YALES2 and the finite volume discretization is presented. The chapter closes with coupling methods used to exchange data between the solvers.

3.2 Discrete element solver

YADE (Yet Another Dynamic Engine) is an open-source DEM solver primarily intended to solve problems arising in granular materials. This code includes a variety of contact laws to describe the interaction physics between particles (spherical, non-spherical and polyhedral shapes). Written primarily in C++, YADE includes Python wrappers for the C++ classes and the user defines a simulation case in a Python script, calling the required ‘engines’. Within the context of this thesis, the contact laws between fiber segments have been modified to include the lubrication terms. Furthermore several C++ and Python functions were written in order to facilitate coupling and data exchange with YALES2.

Figure3.1shows a typical simulation work flow in the DEM solver YADE. Initially the particles with their associated shape, material, translational and rotational velocities and their axis-aligned bounding boxes are initialized. A collision detection algorithm checks for potential contacts via a sweep and prune and insertion sort algorithms [174]. Once a potential contact between a particle pair is identified, geometric parameters that define the ‘indentation depth’ or deformation are calculated. Forces and torques based on the specified contact laws are calculated for each contact pair and stored for the current timestep.

bodies

Shape Material State Bound

interactions

geometry

collision detection pass 2 strain evaluation

physics

properties of new interactions constitutive law

compute forces from strains

forces

(generalized) update bounds collision detection pass 1 other forces (gravity, BC, ...) miscillaneous engines (recorders, ...) reset forces forces → acceleration velocity update

simulation

loop

Figure 3.1: Simulation loop from YADE documentation [174].

For the sake of simplicity, the particles considered in this section are spherical. Acceleration (translational and rotational) of the spheres are calculated from the respective forces and torques. Using a 2nd _{order accurate explicit verlet-velocity integration like scheme, new particle velocities} and positions are calculated. The following set of equations can be written for a sphere. Repre-senting the position of the center of mass of a sphere n as ~On, the translational acceleration at time t can be written as

~¨

On(t) = ~F (t)/mn (3.1)

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3.2 Discrete element solver

equation (3.1), the rotational acceleration ~˙ωn for sphere n with moment of inertia Jn is written as

~˙ωn(t) = ~T (t)/Jn (3.2)

where in equation (3.2), Jn= 2/5mnr2n. In these equations, mn is the mass of the sphere and rn the radius.

For a nonspherical particle, the torque equation (3.2) reads: d

dt(J~˙ωn(t)) = ~T (t) (3.3)

In equation (3.3), J is the moment of inertia tensor.

From equations (3.1), an expression for the position of the particle at t + ∆t can be obtained by Taylor expansion along the particle coordinate ~On(t)as follows:

~ On(t + ∆t) = ~On(t) + ∆t h ~˙ On(t) + (∆t/2) ~¨On(t) i | {z } T1 (3.4) where in equation (3.4), the term T1 is the the Taylor expansion of the velocity ~˙On(t) about ∆t/2, i.e ~˙On(t + ∆t/2). Hence the leapfrog integration equations are

~˙

On(t + ∆t/2) = ~˙On(t− ∆t/2) + ∆tO~¨n(t) (3.5) ~

On(t + ∆t) = ~O(t) + ∆t ~˙On(t + ∆t/2) (3.6) The leapfrog equations for updating the particle orientation is analogous to equations (3.5) and (3.6) for spherical particles. However for non-spherical particles, the numerical integration is quite involved as its local reference frame is not inertial [174]. Details of the rotation integration can be found in ref [174].

3.2.1 Fiber mechanical model

The mechanical model of the fiber in the present work has been previously used to study various problems in geomechanics/geotextiles [34,18,53]. Geometrically, a single fiber segment is rep-resented by a sphero-cylinder, obtained by the Minkowski sum of a sphere and line. Consider a fiber F , as shown in figure8.2, composed of {b1, b2, b3, ..., bn} fiber segments,{c1, c2, c3, ..., cn+1} are the nodes on the fiber segments. { ~O1, ~O2, ~O3, ..., ~On+1} are the position vectors of these nodes and ~Ze is the end to end vector of the fiber . (ex, ey, ez) forms an orthonormal basis that is fixed on the fiber segment and ~p is the orientation vector of a given fiber segment. The length of a fiber segment is defined as l = || ~On− ~On−1||

The fiber segments are massless and the masses are concentrated on the nodes. The equations of motion for a fiber node cn, while neglecting the gravity effects, read as

~ FHYD_cn + ~Fint+ m X i=1 ~ Fcon_cn,ci + m X i=1 ~ Flub_cn,ci = mcnO~¨cn (3.7) ~ THYDcn+ ~Mint+ m X i=1 ~ Mcon_cn,ci = Jbn~˙ωcn (3.8)

where in equation (3.7), ~FHYDcn is the hydrodynamic force, ~Fint is the internal force of the fiber

segment which consists of a normal force component ~FNcn and shear force component ~FScn.

~

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Figure 3.2: Fiber geometry.

force between the fiber segments. The position vector ~O refers to the center of mass of the fiber nodes.

In equation (3.8), ~THYDcn is the hydrodynamic torque, ~Mint is the internal moment consisting

of a bending moment ~MBcn and a twisting moment ~MTcn. Jbn is the inertia tensor, and ~˙ωcn

the angular acceleration. ~Mcon_cn,ci is the moment at the nodes due to the contact forces. The moments are calculated at the point On.

From equations (3.7) and (3.8), the equation of forces and moments on a fiber segment bi having the nodes cnand cn−1 are as follows :

1. The normal force ~FN

~

FNcn = kn(l− lo)~pbi, (3.9)

where lo refers to the length of the fiber segment at the stress free configuration, l the length in the actual configuration and kn is the elastic stiffness associated to the normal force defined as

kn= πEr2_f

l , (3.10)

where in expression (3.10), E is the tensile modulus of the beam material and rf the radius of the fiber segment.

2. The twisting moment ~MT between the nodes

The relative rotation between the nodes is used to define the bending and twisting compo-nents. Using a rotational vector representation, the relative rotation ~Ωn−1,n between the nodes cn−1 and cn is defined as

~

Ωcn−1,cn = ~Ωcn− ~Ωcn−1 (3.11)

The component of twist between the nodes is ~

ΩT_c_n−1_,c_n = (~Ωcn−1,cn. ~pbi). ~pbi (3.12)

and the equation for twisting moment is ~

MT = kt~ΩTcn−1,cn. (3.13)

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kt= πGr_f4

2l , (3.14)

where G is the shear modulus of the beam material. 3. The shear force ~FS and bending moment ~MB

Figure 3.3: Definition of the shear displacement ~us in a fiber segment subjected to pure shear. The bending moment and shear force (figure3.3) are mechanically coupled. Their rates of change are linearly dependent on the rotational and translational velocities of the nodes and can be expressed by introducing a matrix similar to the stiffness matrix of a beam element [193] in structural mechanics [18]

       ~˙ FScn ~˙ MBcn ~˙ FS_cn−1 ~˙ MB_cn−1        =−πEr 4 f 4l3     12 6l ₋₁₂ 6l 6l 4l2 _−6l _2l2 −12 −6l 12 −6l 6l 2l2 _−6l 4l2          ~˙uscn ~ ωBcn ~˙us_cn−1 ~ ωB_cn−1      (3.15)

where in equation (3.15), ~˙uscn is the shear velocity (orthonormal component of the velocity)

of the node defined as

~˙uscn = (I− ~pbi ⊗ ~pbi)·O~˙cn (3.16)

and ~ωBcn is the rotational velocity associated to the bending, defined as

~

ωBcn = ~ωcn− (~ωcn· ~pbi)· ~pbi (3.17)

4. Contact detection between the fiber segments

Consider two non-adjacent (non-parallel, non-intersecting) fiber segments bi and bj with the nodes bicn−1, bicn on bi and bjcn−1,bjcn on bj as shown in figure3.4.

The contact forces between the fiber segments are calculated based on the distance between the segments bi and bj. The direction vector of the segment bi is ~pbi and that of bj is ~pbj.

The position vector of the node bicn is denoted by ~Obicn and the position vector of node

bjcn is denoted by ~Objcn and based on these position vectors the equation of a parametric

line can be written as

~

A(s) = ~Obicn+ s~pbi (3.18)

~

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Figure 3.4: Interaction between non-parallel and non intersecting segments, adapted from [53] where, (s, t) ∈ [0, 1] × [0, 1]. The vector ~w is defined as

~

w(s, t) = ~A(s)_{− ~}B(t) (3.20)

The aim is to find (sc, tc)∈ [0, 1] × [0, 1], such that ~

wc= ~w(sc, tc) = ~A(sc)− ~B(tc) = min

(s,t)∈[0,1]×[0,1]w(s, t)~ (3.21) The vector ~wc = ~w(sc, tc) is uniquely perpendicular to the segment direction vectors ~pbi

and ~pbj. Hence it satisfies : ~pbi. ~wc = 0 and ~pbj. ~wc = 0 and the following system of

equations can be written :

     ~ wc= ~A(sc)− ~B(tc) ~ pbi· ~wc= 0 ~ pbj· ~wc= 0 (3.22) The solution of equation (3.22) is given by

     sc = (~p_bi·~p_bj)(~p_bj· ~w0)−||~p_bj||(~p_bi· ~w0) ||~pbi||||~pbj||−(~pbi·~pbj)2 tc = ||~p_bi||(~p_bj· ~w0)−(~p_bi·~p_bj)(~p_bi· ~w0) ||~p_bi||||~p_bj||−(~p_bi·~p_bj)2 (3.23) where in equation (3.23), ~wo = ~Obicn−1− ~Objcn−1.

Once the values of sc and tc are obtained, the criterion for the contact is defined as: || ~wc(sc, tc)|| ≤ ||sc− tc|| (3.24) 5. Contact force between two fiber segments

The contact between two fiber segments bi, bj having radii rbi, rbj is shown in figure 3.5.

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Figure 3.5: Interaction between two fiber-segments

the contact is associated with one virtual node at the contact points scand tc. The virtual nodes Si and Sj are positioned along the vectors connecting the nodes of beam segments bi, bj in contact as shown in figure 3.5. ||~wc(sc, tc)|| = ||~Si − ~Sj|| defines the shortest distance between the respective axes of the segments in contact. The translational ~˙Si and rotational velocities ~Sωi of a virtual node Si are calculated from its associated segment in

the following way[53,18]:

~˙Si= λ ~˙Ocn−1+ (1− λ)O~˙cn. (3.25) ~ Sωi = λ~ωcn−1+ (1− λ)~ωcn. (3.26) λ = || ~Ocn−1 − ~Si|| || ~Ocn−1− ~Ocn|| . (3.27)

The contact force ~Fcon consists of a normal ~Fcon,N and shear component ~Fcon,S, the ex-pression for the normal force reads as

~

Fcon,N = knconun,con· ~ncon (3.28)

where un,con is the normal displacement (indentation depth) and ~ncon the contact normal between the virtual nodes, which are defined as

un,con =||~Si− ~Sj|| − rSi− rSj (3.29a) ~ncon = ~ Si− ~Sj ||~Si− ~Sj|| (3.29b) where in equation (3.29a) rSi and rSj are the radii of the virtual nodes and rSi = rbi,

rSj = rbj.

kncon is the normal stiffness between the contact nodes

kncon = 2ˆk

rSirSj

rSi + rSj

(3.30a) where in equation (3.30a), ˆk is the normalized stiffness and ˆk is assigned the value of E. The shear stiffness kscon is defined as

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where in equation (3.30b), α is a non dimensional parameter. The contact shear force ~Fcon,S is as follows:

~

F_con,St+dt = ~F_con,St + kscon~˙uscondt (3.31a)

|| ~Fcon,S|| ≤ µf|| ~Fcon,N|| (3.31b) where in equation (3.31a), ~˙uscon is the relative shear velocity defined as

~˙uscon = (I− ~nncon⊗ ~nncon)·

~˙Si−~˙Sj + ~Sωi− ~Sωj × S~i− ~Sj 2 ! (3.32) and µf is the coefficient of friction. Equation (3.31) leads to an elastic-frictional condition between the fiber segments in contact.

Hence the expression for the contact force reads as ~

Fcon= ~Fcon,N + ~Fcon,S (3.33)

The moment of this force about Si is: ~

Mcon= ~C− ~Si

× ~Fcon (3.34)

considering the contact point located at ~C = (~Si+ ~Sj)/2

The contact force ~Fcon and the corresponding moment ~Mcon are then distributed to the nodes of the segment using λ as the interpolation weight. Further details are reported in the works of Bourrier et al. [18] and Effeindzourou et al. [53,174]

6. Lubrication forces

The lubrication force between the fiber segments is based on the formulation of Frenkel and Acrivos [59], and it is composed of a normal component and a shear component.The normal component of the lubrication force is written as

~ F_nL= 3

2πµ r2_f

h~˙un. (3.35)

The equation for the shear component is ~ F_sL= πµ 2 − 2rf+ (2rf + h) + ln (2rf + h) h ~˙us, (3.36)

where in equations (3.35) and (3.36), µ is the fluid viscosity, rf is the radius of the fiber segment, h is the distance between 2 adjacent fibers (calculated based on equation (3.22)), ~˙un and ~˙us are the normal and shear velocity components of the fiber segment.

Numerical modeling of long flexible fibers in inertial flows.

HAL Id: tel-01877792

https://tel.archives-ouvertes.fr/tel-01877792

Numerical modeling of long flexible fibers in inertial

flows.

Deepak Kunhappan

To cite this version:

DOCTEUR DE LA COMMUNAUTE UNIVERSITE

GRENOBLE ALPES

KUNHAPPAN Deepak

Modélisation numérique de

l'écoulement suspensions de

fibres souples en régime inertiel

Numerical modeling of long flexible fibers

in inertial flows

Abstract

Résumé

Acknowledgements

CONTENTS

CHAPTER

1

INTRODUCTION

Introduction

Objectives

Thesis outline

CHAPTER

2

REVIEW OF COMPUTATIONAL METHODS FOR PARTICULATE

MULTIPHASE FLOWS

2.1

Introduction

2.2

Microhydrodynamics

2.3

Computational modelling of particulate multiphase flows

Resolved

Resolved

Modeled

Resolved

Modeled

2.4

Numerical modeling of rigid and flexible fibers

2.5

Objectives of the present work

CHAPTER

3

NUMERICS AND COUPLING METHODOLOGY

3.1

Introduction

3.2

Discrete element solver

bodies

interactions

forces

simulation

loop

_Modeled